Hong Life CalculatorsUnderstanding the RL time constant calculator — concept and principle
This calculator gives the time constant of a resistor–inductor circuit and the current flowing at a chosen moment after a DC voltage is switched on, together with the final steady current. It is used for relay and solenoid coils, motor windings and any coil whose current rise time matters.
How it is calculated
- Time constant: τ = L / R. With L in millihenries and R in ohms, τ comes out directly in milliseconds.
- Current at time t: i(t) = (V/R) · (1 − e^(−t/τ)), with t in milliseconds.
- Final current: I_final = V / R, reached once the inductor stops opposing the current.
The principle
When the switch closes, the inductor produces a back-voltage L · di/dt that opposes any change in current. At the first instant the current is zero and the whole supply voltage appears across the coil. As the current grows, more of the voltage drops across the resistor (iR) and less is left to drive the change, so the rate of rise slows. The circuit equation V = iR + L · di/dt has an exponential solution, the mirror image of a charging capacitor. The current covers 63.2% of the way to V/R in one τ, 86.5% in 2τ, 95.0% in 3τ and 99.3% in 5τ. A larger inductance stores more energy and therefore takes longer; a larger resistance limits the final current and shortens τ.
Worked example
- Inputs: R = 100 Ω, L = 50 mH, V = 12 V, t = 0.5 ms.
- τ = 50 ÷ 100 = 0.5 ms.
- Final current: 12 ÷ 100 = 0.12 A.
- At t = 0.5 ms, t/τ = 1, so i = 0.12 × (1 − e^(−1)) = 0.12 × 0.632121 = 0.075854 A (about 75.9 mA).
| Resistance (Ω) | τ = L/R (ms) |
|---|---|
| 25 | 2 |
| 50 | 1 |
| 75 | 0.6666666667 |
| 100 | 0.5 |
| 125 | 0.4 |
| 150 | 0.3333333333 |
| 200 | 0.25 |
How to read the result
If t is a small fraction of τ, the current has barely started; after about 5τ the coil behaves like a plain resistor (its winding resistance). For a relay, compare the current at your time of interest with the pull-in current on the datasheet to see how long the contact takes to operate. A short τ means a fast response; a long τ smooths current ripple, which is why chokes are used in power supplies.
Common mistakes
- Mixing units: L in henries with t in milliseconds gives a result 1000 times off. Keep L in mH and t in ms as the fields expect.
- Leaving out the coil's own winding resistance and the source resistance; R must be the total series resistance.
- Opening the circuit suddenly. Current in an inductor cannot change instantly, so the coil produces a large voltage spike unless a flyback diode or snubber gives the current a path.
Frequently asked questions
What happens when the voltage is removed?
If a path such as a diode is provided, the current decays as i = I₀ · e^(−t/τ) with the same τ, where τ is set by the resistance of that decay path.
How is RL different from RC?
In RC the capacitor voltage cannot jump; in RL the inductor current cannot jump. Both follow the same exponential curve with one time constant.
Sources: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, McGraw-Hill, chapter on first-order circuits · D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, Wiley, chapter on inductance